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True, but what's puzzling me about the moon is that (contrary to our planet) it always has the same hemisphere facing Earth, and this confuses me a little bit about what's really going on. But I must confess I completely neglected the sun in my analysis. Maybe Solar tides can save the day after all...

I missed it too. I tried to write a couple times, but didn't produce anything more than disjointed paragraphs. I wrote a story yesterday that I want to post, but it needs some editing first.

Oh yeah, the thing with the moon not rotating, I forgot about that. oops I guess only solar tides then.

if you just mean the moon is going faster around the earth I think there wouldn't be much of a difference. The earth's atmosphere keeps us closed in (the way you can be closed within a moving car) such that we don't feel its movement (although we would if it suddenly stopped or changed pace)

the water would stay as normal

the tides would still be affected by the earth and the sun. If the earth is between the moon and the sun then it would be high tide on the moon's earth-facing face. If the moon is between the earth and the sun then the tides would be at their lowest on that face

If the earth is between them then the tides would be lowest on the out-facing face, and if the moon is between the earth and the sun then the tides would be highest on the out-facing face

basically the water would always lean towards the earth, but would bend a bit towards the sun too

also the tides would be much stronger than on earth. The moon has a much weaker gravitational pull, and the earth has a much stronger

I think the problem of the magnitude of ocean tides on the Moon is a complicated one. Since the moon is tidally locked to the Earth, the Earth won't directly influence lunar tides to the extent the Moon does to tides on Earth. Lunar tides will be determined by the changing relative positions of the Earth, Moon, and Sun, with the Sun being the driver of daily tidal variations. This is significantly different than the case on Earth, where the Moon, a nearby lesser gravity object, contributes the bulk of the gravity gradient resulting in tides. The Sun has a huge gravitational potential, but, due to its distance, a much smaller gravitational gradient at the position of the Earth. The Moon's small size further reduces the impact of the sun, however the extent of its orbit around the Earth increases it. The remaining consideration would be the extent of any liquid bodies on the Moon, as that determines whether tidal effects are of any significance.

Given that this is a classical problem involving well known objects I'm guessing someone in over the past four centuries has worked it out in excruciating detail, but it's little far afield from my focus for me to easily find it at the moment.

I like how I clicked on a thread about NaNoWriMo and started reading people debating how tides would work if the moon had oceans. Anyway, I see Onnes replied, so I probably don't need to give any input. I'll just add that the basic principle behind tides is that you stop treating the bodies in the gravitational system as point masses, so now you've got to deal with how the half of the moon facing the Earth is closer and hence is feeling a stronger force from the Earth than the other half, and the half of the Earth facing the moon is also producing stronger force than the back half of the Earth, etc.. So the halves you're talking about change because the bodies are rotating (and yes, the moon is rotating: it goes around once on its axis in exactly the same time it takes for it to orbit the Earth once, hence why you always see the same side).
So I don't know. Start with that, add the sun, and think it through and you'll have your answer.

Anyway, on topic, I'm doing NaNo for the second time this year (last time I guess was in 2009). So far what I've written just might be the worst thing I've ever produced. I'll finish, I'm sure, but I may never actually disclose this project to the world at large, at least not without some heavy revisions.